Minimization Problem - rowing and walking

Minimizing Time it Takes to Row and Walk to a Given Point

 

We're in a rowboat two miles from shore on a cold, rainy river. There's a tent and a nice campfire three miles down the river, and one mile inland. We can row at two miles an hour, and walk at four miles an hour. We need to find the spot on the shore that will get us to camp in the shortest time.

First, as always, draw a picture.

 

We ask:

1.       What do we want to do?

We want to get to camp as fast as possible. In math terms, that means we want to minimize the time it takes to get from where we are to camp.

2.       How do we accomplish this?

We find the derivative of the equation for time, and set it to zero.

3.       What formulas can help us?

Time = distance / rate

Distance = sqrt(a^2 + b^2)

4.       What do we know already?

The time it will take to row to the spot x miles down the river is sqrt(x^2 + 4)/2.

The time it will take to walk from point x to camp is sqrt((3 –x)^2 + 1)/4

 

5.       What is the problem we need to solve?

We need the derivative of time. We set up the equation for total time we need:

 

T = sqrt(x^2 + 4)/2 + sqrt(x^2 + 4)/2

We find the derivative of the time equation:

dT = x/ (2sqrt(x^2 + 4))  -  (3 –x)/ (4sqrt( (3 –x)^2 + 1))

 

 

We set dT to zero:

 

x  (2sqrt(x^2 + 4))  -  (3 –x) / (4sqrt( (3 –x)^2 + 1) ) = 0

 

x / ( 2sqrt(x^2 + 4) )  =  (3 –x) / (4sqrt( (3 –x)^2 + 1) )

 

 

At this point, the algebra gets a tiny bit intricate. Go slow. I got into trouble by rushing.

We cross-multiply, and eventually we end up with:

 

12x^4 – 72x^3 + 108x^2 + 96x – 144 = 0

 

We can divide the whole thing by 12, ending up with:

 

x^4 – 6x^3 + 9x^2 + 8x – 12 = 0

 

This is where I love Synthetic Division. We know that the only possible integer answers are 0, 1, 2, and 3. My policy is to always try 1 first in Synthetic Division.

 

1 is the answer. Rowing to a point 1 mile down the river will get us to camp quickest.

In math terms, x = 1 will minimize the value of time (T).

 

Since we know that x has to be between 0 and 3, we can always plug numbers in that range into the derivative and see which number makes the derivative equal to zero. It works easily in this case because the answer is an integer. But that's cheating, kind of, so you didn't hear it from me.